In asymptotically flat space-time a very interesting connection exists [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]} between soft-theorems and Ward identities of infinite dimensional (asymptotic) global symmetries. In even space-time dimensions conformal techniques are useful for studying this correspondence. In this subject, one basic fact is that the Lorentz group \(SO(D-1,1)\)  acts on the celestial sphere Celestial sphere of a point in Minkowski space is essentially the space of null directions at that point. This space is topologically a sphere. \(S^{D-2}\)  as the conformal group. In fact, there are interesting operators in the theory, called soft operators [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [24]}, [25]}, which live on the celestial sphere and transform as conformal primary under the (Lorentz) conformal group. It has also been conjectured [26]}, [24]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [25]}, [35]}, [36]}, [37]}, [38]}, [39]}, [40]} that there is an Euclidean CFT on the celestial sphere which holographically computes the \(S\) –matrix in asymptotically flat space-time. Motivated by this conjecture, we try to understand in this paper the interplay between the conformal field theory structure and the infinite dimensional global symmetries. One way to get some hints of this interplay is to try to derive the soft-theorems from Ward-identities in a pure S–matrix theory. By "pure \(S\) –matrix" we mean an asymptotic description. Therefore bulk equations of motion – classical or quantum – or the concept of bulk gauge symmetry are not naturally defined in this context. It seems that such an approach may not be completely unrealistic because we now have infinite dimensional global symmetries which act on the space of \(S\) –matrix elements. Moreover, these global symmetries are also asymptotic symmetries [1]}, [2]}, [43]} and so, roughly speaking, constitute the subset of the bulk gauge transformations which act on the physical Hilbert space. So at least in principle, nothing should be lost if we focus only on this physical part. Let us now briefly describe how we approach this problem.
